Here is some background on my question. The Weierstraß function (as well as its translates) $\wp(x)$ solves the implicit first-order ODE $$(y')^2=4y^3-g_2y-g_3.$$ Differentiating gives the second-order explicit ODE $$y''=12y^2-g_2, $$ in which $g_3$ makes no appearance.

$\wp$ can be used to express the solution of any first-order equation of the form $$(y')^2=a_3 y^3+a_2 y^2+a_1 y+a_0, $$ or equivalently, any second-order equation of the form $$y''=b_2 y^2+b_1 y+b_0. $$ In both cases, all it takes is a simple change of variables to remove one of the powers, and get the leading coefficient right.

My question is about differential equations whose RHS is one degree larger, that is $$(y')^2=a_4 y^4+a_3y^3+a_2y^2+a_1 y+a_0 , \tag{1}$$ or $$y''=b_3 y^3+b_2y^2+b_1 y+b_0. \tag{2} $$

In some special cases, such as $$y''=2k^2 y^3-(1+k^2) y, $$ and $$y''=-2y^3+(2-k^2)y,$$ the Jacobi elliptic functions $\operatorname{sn},\operatorname{dn}$ play a role in the solution.

Here is my question: For which values of the coefficients in $(1)$ and $(2)$ can the ODEs be solved explicitly in terms of special functions? What are the solutions in those cases?

Thank you!


The change of the function $y= \frac{\alpha z+\beta}{\gamma z+\delta}$ with $\alpha\delta-\beta\gamma=1$ transforms (1) into $$\left(z'\right)^2=a_4\left(\alpha z+\beta\right)^4+ a_3\left(\alpha z+\beta\right)^3\left(\gamma z+\delta\right) +a_2\left(\alpha z+\beta\right)^2\left(\gamma z+\delta\right)^2+ +a_1\left(\alpha z+\beta\right)\left(\gamma z+\delta\right)^3+ a_0\left(\gamma z+\delta\right)^4=\operatorname{poly}_4\left( z\right).$$ We can make the coefficient of $z^4$ disappear by the appropriate choice of $\alpha/\gamma$. The problem is thus completely equivalent to the previous differential equation with cubic right hand side and can be solved in terms of Weierstrass $\wp$-function.

The change of variable above is exactly the one which transforms the integral $\int\frac{dy}{\sqrt{\operatorname{poly}_4\left( y\right)}}$ into $\int\frac{dz}{\sqrt{\operatorname{poly}_3\left( z\right)}}$. This has the following geometric interpretation: $w^2=\operatorname{poly}_4\left( y\right)$ is an elliptic curve realized as a $2$-sheeted covering of the Riemann sphere having $4$ branch points. Cubic Weierstrass form of the curve corresponds to moving one of the branch points to $\infty$ by a homography.


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